President Trump announced a deal to replace NAFTA on Oct. 1. To the right is his chief negotiator, Robert Lighthizer. During negotiations, Trump personally insulted Canadian Prime Minister Justin Trudeau with words that stung many Canadians.

Shortly after Canada agreed to new terms for the North American Free Trade Agreement, the Trump administration held up the deal as vindication of the president’s bare-knuckles negotiating style.

And the administration was certainly right about the tactics: During 13 months of talks, Trump imposed hefty tariffs on Canadian and Mexican steel, threatened new taxes on the two nations’ automotive exports, repeatedly warned of blowing up NAFTA and made disparaging personal remarks about Canada’s prime minister.

“To me, the fact that we got a good agreement is all the evidence you need that the president’s approach is a good approach; it’s the right approach,” Robert Lighthizer, Trump’s chief negotiator, said when the new NAFTA, renamed the United States-Mexico-Canada Agreement, was announced last week.

But whatever the gains turn out to be for American businesses and workers — and they will be hotly debated before Congress votes on the new agreement, probably next year — the costs of Trump’s hard-nosed diplomacy look to be considerable in crucial areas outside the field of financial returns.

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The president’s bruising strategy led to frayed relations with some of the United States’ closest allies and contributed to what has become a stain on America’s image and that of its president across the globe.

The consequences of declining goodwill may not be immediately apparent, stretching beyond Trump’s tenure in the White House. But they could potentially end up overshadowing whatever immediate profits his hardball negotiating style yields.

“The U.S. has been a global leader for 70 years, but the cost of going to a narrow balance-sheet definition of relationship is that it is no longer seen as a leader,” Eric Miller, a global fellow at the Wilson Center’s Canada Institute in Washington, said.

“Leadership is about inspiring followers,” he said. “What’s happening over time with “America first” is it’s becoming America alone.”

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Jeffrey Engel, director of the Center for Presidential History at Southern Methodist University, said, “Trump, by his own admission, is a transactional leader” — that is, one who sees issues in terms of deals, in which one side tries to gain advantages over the other.

Because of the United States’ economic and political heft, Engel said, “if you’re a transactional president and bluster and take every small and big negotiation to the hilt, you can win.”

But the price of those financial victories, Engel said, is that America’s allies and partners may not be there when the United States needs them on other important issues, such as counterterrorism, pressuring Iran or cooperating with the Pentagon and on crucial intelligence operations.

“American presidents have found their popularity abroad extraordinarily useful. ... It’s enabled them to parlay that into diplomatic leverage,” Engel said, citing as one example the broad support the United States and President George H.W. Bush got during the 1990-91 Gulf War.

Trump, instead of maintaining or building on the relationships forged by presidents dating back to the end of World War II, has almost gone out of his way to alienate allies.

In part this is because Trump views foreign relationships as primarily matters of potential financial gain or loss for American business. He has reportedly groused repeatedly, for example, that Pentagon generals ignore the moneymaking potential in dealings overseas. He has complained that the United States did not seize Iraqi oil after the 2003 Iraq war.

In addition, Trump has found that an in-your-face approach to foreign leaders and portraying them as taking advantage of America are valuable tools for firing up his political base.

Accordingly, Trump has accused North Atlantic Treaty Organization partners of not contributing enough money for the common defense. He has pulled out of the international Paris climate accord and the multinational nuclear agreement with Iran. All the while, he has gone after the European Union and Japan on trade, pressuring them to make concessions or face stiff auto tariffs.

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The result has been a widespread souring of the way other countries and their citizens view the United States.

Internationally, public opinion of Trump is at such historical lows for an American president that even Russian leader Vladimir Putin and China’s Xi Jinping rated higher, on average, in a recent global survey by Pew Research released this month.

Pew’s study did not include Chinese citizens, but by all indications, the Chinese public’s attitude toward Trump also has hardened in recent months as his administration has escalated its trade battle.

Loss of confidence in the U.S. president and America’s world leadership has been especially steep in Mexico, Japan and across Western Europe.

No country, however, has felt as bruised as Canada. It wasn’t just that Trump slapped 25% tariffs on Canadian steel and aluminum on the grounds that its production constituted a national security threat to the United States — something inconceivable to many Canadians.

To top things off, as NAFTA negotiations were heating up this summer, the U.S. side threatened to levy large tariffs on Canadian auto imports, which many regarded as potentially crippling to Canada’s economy.

Many Canadians had believed they had a profound and unshakable bond with their much bigger southern neighbor, with long, open borders between them, deep economic integration that predated the 24-year-old NAFTA and a military alliance that was as close as brothers.

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“It’s difficult to convey how shocking this was to Canadians,” said Janice Stein, founding director of the Munk School of Global Affairs at the University of Toronto. She noted that no U.S. president had ever made such threats to bring down the Canadian economy or publicly maligned Canada’s leaders.

Trump, for his part, in remarks after Canada and the United States struck a deal on a new NAFTA, acknowledged the “strong tensions” during negotiations but suggested that bilateral ties, as well as his personal relations with Trudeau, would get right back on track.

“We have a great relationship,” Trump said, praising Trudeau as a “good man” who loves his country.

But most experts doubt the insults and threats will be forgotten quickly. “Trust-building is a very long process, and destroying it is very fast,” Stein said. “That’s why people don’t use this strategy.”

One certain result, she said, is that Canada will make a concentrated effort to diversify its economy, now heavily dependent on the United States, by building trading and investment relationships with others.

Anecdotal reports suggest Canadian tourism to the United States is also likely to slip. “I think Florida will see less Canadians this year,” Stein said.

And experts said Canadian regulators may be a little more wary of approving an American firm trying to make an acquisition of a Canadian company.

Beyond Canada and Mexico, Richard Wike, Pew’s director of global attitudes research, said: “Public opinion about the United States is an important contextual factor. It helps set the incentive structure for politicians around the world, making it easier or harder to get cooperation.”

That could prove challenging for the Trump administration as it looks for allies to join in its trade fight with China.

Now that NAFTA has been largely settled, the White House is turning its focus on Japan and the EU, but ultimately it sees China as its primary target on trade and broader bilateral relations.

“The risk is that by creating economic friction with allies, they might not get the strong support on China,” said Mireya Solís, chair of Japan studies at the Brookings Institution. That could be a side effect even though Japan shares America’s concerns about China’s mercantilist behavior, including theft of intellectual property and forced technology transfer.

Japan agreed to come to the table with the United States only because Japanese officials and business leaders were afraid of Trump’s auto tariffs. And although negotiations aren’t expected until next year, Solís said the damage already has been done.

“I think a very dangerous precedent has been set … that the president and his team think that using tariffs and the threat of tariffs is the way to get trade agreements with provisions they like,” she said.

Indeed, the Trump administration’s Lighthizer made clear that NAFTA would serve as a template for future trade agreements, which he said would seek elimination of unfair trading practices.

He credited Trump for his “leadership, vision and grit” in pushing through this new paradigm-shifting model on trade.

Miller, of the Wilson Center, agreed it took an unusual American president who “wasn’t just willing to walk away but willing to drop tariff nuclear bombs on the auto industry” to rewrite NAFTA and get as much as he did from trading partners. These include tighter rules on autos and auto parts and more access to Canada’s politically sensitive dairy market.

Still, Miller wondered whether those gains were worth it, given the price to be paid in other areas.

Don Lee covers the U.S. and global economy out of Washington, D.C. Since joining the Los Angeles Times in 1992, he has served as the Shanghai bureau chief and in various editing and reporting roles in California. He is a native of Seoul, Korea, and graduated from the University of Chicago.